Arithmetic properties of pullback domains.

摘要

Let T be an integral domain, I an ideal of T, D a domain contained in E = T/I, and R the domain defined by the following pullback of canonical homomorphisms: R&rarrr;D ↓↓ T-→4 T/I=E. Thus R = ϕ-1(D). Many authors have studied arithmetic properties in such diagrams, but, until now, definitive results have required that I be a maximal ideal of T or that T be a valuation domain. Our purpose is to eliminate, insofar as is possible, these extra hypotheses. For example, we prove, in the notation above, that R is a Prufer domain if and only if D and T are Prufer domains, I is a prime ideal of T , and D and E have the same quotient fields. We obtain a similar result for the Bezout property. More importantly, with mild restrictions on T and I, we characterize when R is a Prufer v-multiplication domain or a GCD domain; this requires substantial new techniques, including a generalization of the notion of star operations.